How do you find the derivative of #(cos^2(t) + 1)/(cos^2(t))#?

1 Answer
Bio
Mar 30, 2016

#frac{"d"}{"d"t}( frac{cos^2(t)+1}{cos^2(t)} ) = frac{2sin(t)}{cos^3(t)}#

Explanation:

Try to simplify the expression.

#frac{cos^2(t)+1}{cos^2(t)} = 1 + sec^2(t)#

Next, let #u = sec(t)#.

#frac{"d"u}{"d"t} = sec(t)tan(t)#

So,

# 1 + sec^2(t) = 1 + u^2#

Now, differentiate using the chain rule.

#frac{"d"}{"d"t}(1+u^2) = frac{"d"}{"d"u}(1+u^2) * frac{"d"u}{"d"t}#

#= 2u * sec(t)tan(t)#

#= 2sec(t) * sec(t)tan(t)#

#= 2sec^2(t)tan(t)#

#= frac{2sin(t)}{cos^3(t)}#

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